A Note on Rectangular Quotients

1

• A Note on Rectangular Quotients
• By
• Achiya Dax
• Hydrological Service
• Jerusalem , Israel
• e-mail dax20_at_water.gov.il

2

• The Symmetric Case
• S ( sij ) a symmetric positive
  semi-definite n x n matrix
• With eigenvalues l1 ³ l2 ³ ... ³
  ln ³ 0
• and eigenvectors v1 , v2 , ,
  vn
• S vj lj vj , j 1, , n . S V
  V D
• V v1 , v2 , , vn , VT V V VT I
• D diag l1 , l2 , , ln
  
• S V D VT S lj vj vjT

3

• Low-Rank Approximations
• S l1v1v1T lnvnvnT
• T1 l1v1v1T
• T2 l1v1v1T l2v2v2T
• .
• .
• .
• Tk l1v1v1T l2v2v2T lkvkvkT
• Tk is a low - rank approximation of
  order k .

4

• The Rayleigh Quotient
• r r (v , S) vT S v / vTv
• r arg min f ( q ) S v - q v 2
  
• r estimates an eigenvalue corresponding
  to V

5

• The Power Method
• Starting with some unit vector p0 .
• The k th iteration, k 1, 2, 3, ,
• Step 1 Compute wk S pk-1
• Step 2 Compute rk (pk-1)T wk
• Step 3 Normalize pk wk / wk 2
  

6

• THE POWER METHOD
• Asymptotic Rates of Convergence
• ( Assuming l1 gt l2 )
• pk a v1 at a linear rate, proportional to
  l2 / l1
• rk a l1 at a linear rate, proportional to
  (l2 /l1)2
• Monotony l1 ³ ³ rk ³ ³ r2 ³ r1 gt 0

7

• THE POWER METHOD
• The asymptotic rates of convergence
• depend on the ratio l2 / l1
• and can be arbitrarily slow.
• Yet rk provides a fair estimate of l1
• within a few iterations !
• For a worst case analysis see
• D.P. OLeary, G.W. Stewart and J.S.
  Vandergraft, Estimating the largest eigenvalue
• of a positive definite matrix, Math. of Comp.,
  33(1979), pp. 1289 1292.

8

• THE POWER METHOD
• An eigenvector vj is called
• large if lj ³ l1 / 2 and small if
  lj lt l1 / 2 .
• In most of the practical situations,
• for small eigenvectors pkT vj becomes
  negligible
• after a small number of iterations.
• Thus, after a few iterations pk actually lies
  
• in a subspace spanned by large
  eigenvectors.

9

• Deflation by Subtraction
• S l1 v1 v1T ln vn vnT .
• S1 S - l1 v1 v1T l2 v2 v2T ln vn
  vnT .
• S2 S1 - l2 v2 v2T l3 v3 v3T ln
  vn vnT .
• .
• .
• .
• Sn-1 ln vn vnT .
• Sn 0 .
• 
  Hotelling (1933, 1943)

10

• The Frobenius norm
• A ( aij ) , A F S S aij
  2½

11

• The Minimum Norm Approach
• Let the vector v solve the minimum norm
  problem
• minimize E (v) S - v vT
  F 2 .
• Then
• v1 v / v 2 and l1
  (v)T v .

12

• The Symmetric Quotient
• Given any vector u , the Symmetric
  Quotient
• g (u) uT S u / (
  uT u ) 2
• solves the one parameter problem
• minimize f (q) S - q u uT
  F 2
• That is,
• g (u) arg min
  f (q) .
• If u 2 1 then g
  (u) r (u) uT S u

13

• The Symmetric Quotient Equality
• S - g(u) u uT F 2 S
  F 2 - ( r(u) ) 2
• means that solving
• minimize F (u) S - u uT F 2
• is equivalent to solving
• maximize r (u ) uT S u / uTu

14

• Can we extend these tools
• to rectangular matrices?

15

• The Rectangular Case
• A ( aij ) a real m x n matrix , p
  min m , n
• With singular values s1 ³ s2 ³ ³ sp
  ³ 0 ,
• Left singular vectors u1 , u2 ,
  , up
• Right singular vectors v1 , v2 ,
  , vp
• A vj sj uj , AT uj sj v
  j 1, , p .

16

• The Singular Value Decomposition
• A U S V T
• S diag s1 , s2 , , sp , p
  min m , n
• U u1 , u2 , , up , UT U
  I
• V v1 , v2 , , vp , VT V
  I
• A V U S AT U V S
• A vj sj uj , AT uj sj vj j
  1, , p .

17

• Low - Rank Approximations
• A U S VT S sj uj
  vjT
• A s1 u1 v1T s2 u2 v2T sp
  up vpT .
• B1 s1 u1 v1T
• B2 s1 u1 v1T s2 u2 v2T
  
• .
• .
• .
• Bk s1 u1 v1T s2 u2 v2T sk
  uk vkT
• Bk is a low - rank approximation of
  order k .
• (Also called "truncated SVD or filtered
  SVD.)

18

• The Minimum Norm Approach
• Let the vectors u and v solve the
  problem
• minimize F ( u , v) A - u vT
  F2
• then
• u1 u / u 2 , v1 v /
  v 2 ,
• and
• s1 u 2 v
  2
• ( See the Eckhart-Young, Schmidt-Mirsky
  Theorems.)

19

• The Rectangular Quotient
• Given any vectors , u and v ,
• the Rectangular Quotient
• h (u , v) uT A v / ( uT u ) (
  vT v )
• solves the one parameter problem
• minimize f (q) A - q u vT
  F 2
• That is,
• h (u , v) arg min f
  (q)

20

• The Rectangular Rayleigh Quotient
• Given two vectors , u and v ,
• the Rectangular Rayleigh Quotient
• r(u , v) uT A v / u 2 v
  2
• estimates the corresponding singular value.

21

• The Rectangular Rayleigh Quotient
• Given two unit vectors , u and v ,
• the Rectangular Rayleigh Quotient
• r(u , v) uT A v /
  u 2 v 2
• solves the following three problems
• minimize f1(q) A - q
  u vT F
• minimize f2(q) A v -
  q u 2
• minimize f3(q) AT u -
  q v 2

22

• The Rectangular Quotients Equality
• Given any pair of vectors, u and v , the
• Rectangular Quotient
• h (u ,v) uT A v / ( uT u ) ( vT
  v )
• satisfies
• A h (u ,v) u vT F 2 A F 2 -
  ( r(u ,v) ) 2

23

• The Rectangular Quotients Equality
• Solving the least norm problem
• minimize F ( u , v) A - u vT
  F 2
• is equivalent to solving
• maximizing r(u , v) uT A v / u 2
  v 2

24

• Approximating a left singular vector
• Given a right singular vector , v1 , the
  corresponding
• left singular vector , u1 , is attained by
  solving
• the least norm problem
• minimize g ( u ) A - u v1T F
  2
• That is,
• u1 A v1 / v1T v1 .
• ( The rows of A are orthogonalized against
  v1T .)

25

• Approximating a right singular vector
• Given a left singular vector , u1 , the
  corresponding
• right singular vector , v1 , is attained by
  solving
• the least norm problem
• minimize h ( v ) A u1 vT F
  2
• That is,
• v1 AT u1 / u1T u1 .
• (The columns of A are orthogonalized
  against u1 .)

26

• Rectangular Iterations - Motivation
• The k th iteration , k 1, 2, 3, ,
• starts with uk-1 and vk-1 and ends
  with uk and vk .
• Given vk-1 the vector uk is obtained by
  solving the problem
• minimize
  g(u) A - u vk-1T F 2 .
• That is,
• 
  uk A vk-1 / vk-1T vk-1 .
• Then , vk is obtained by solving the problem
• minimize
  h(v) A - uk vT F 2 ,
• which gives
• 
  vk AT uk / ukT uk .

27

• Rectangular Iterations Implementation
• The k th iteration , k 1, 2, 3, ,
• uk A vk-1 / vk-1T vk-1 ,
• vk AT uk / ukT uk .
• The sequence vk / vk 2 is obtained
  by applying
• the Power Method on the matrix ATA .
• The sequence uk / uk 2 is obtained
  by applying
• the Power Method on the matrix AAT .

28

• Left Iterations
• uk A vk-1 / vk-1T vk-1 ,
• vk AT uk / ukT uk .
• ---------------------
  --------------------------------------------------
  --------------------------------
• vkT vk vkTAT uk / ukT uk
• Right Iterations
• vk AT uk-1 / uk-1T
  uk-1 ,
• uk A vk / vkT vk .
• -------------------
  --------------------------------------------------
  ---------------------------------
• ukT uk ukTA vk / vkT
  vk
• Can one see a difference?

29

• Some Useful Relations
• In both cases we have
• ukT uk vkT vk ukTA vk ,
• uk 2 vk 2 ukT A vk / uk 2
  vk 2 r(uk , vk) ,
• and h(uk , vk) ukT A vk / ukT uk
  vkTvk 1 .
• The objective function F ( u , v ) A -
  u vT F 2
• satisfies F ( uk , vk) A F
  2 - ukT uk vkT vk
• and F( uk , vk) - F( uk1 , vk1)
• uk1T uk1 vk1T vk1 - ukT uk vkT
  vk gt 0

30

• Convergence Properties
• Inherited from the Power Method , assuming
  s1 gt s2 .
• The sequences uk / uk 2 and vk
  / vk 2
• converge at a linear rate, proportional to
  (s2 / s1 ) 2 .
• ukT uk vkT vk a ( s1 )
  2
• at a linear rate, proportional to (s2 / s1
  ) 4
• Monotony
• ( s1 )2 ³ uk1T uk1 vk1T vk1 ³ ukT uk
  vkT vk gt 0

31

• Convergence Properties
• rk uk 2 vk
  2
• provides a fair estimate of s1
• within a few rectangular iterations !

32

• Convergence Properties
• After a few rectangular iterations
• rk , uk , vk
• provides a fair estimate of a
• dominant triplet
• r1 , u1 , v1
  .

33

• Deflation by Subtraction
• A1 A s1 u1 v1T sp up vpT .
• A2 A1 - s1 u1 v1T s2 u2 v2T sp up
  vpT
• A3 A2 - s2 u2 v2T s3 u3 v3T sp vp
  vpT
• .
• .
• .
• Ak1 Ak - sk uk vkT sk1 uk1 vk1T
  spup vpT
• .
• .
• .

34

• Deflation by Subtraction
• A1 A
• A2 A1 - s1 u1 v1T
• A3 A2 - s2 u2 v2T
• .
• .
• .
• Ak1 Ak - sk uk vkT
• .
• .
• .
• where sk , uk , vk denotes a computed
• dominant singular triplet of Ak .

35

• The Main Motivation
• At the k th stage , k 1, 2, ,
• a few rectangular iterations
• provide a fair estimate of
• a dominant triplet of AK .

36

• Low - Rank Approximation Via Deflation
• s1 ³ s2 ³ ³ sp ³ 0 ,
• A s1 u1 v1T s2 u2 v2T sp up
  vpT .
• B1 s1 u1 v1T ( means computed
  values )
• B2 s1 u1 v1T s2 u2 v2T
  
• .
• .
• .
• Bl s1 u1 v1T s2 u2 v2T sl
  ul vlT
• Bl is a low - rank approximation of
  order l .
• ( Also called "truncated SVD or the
  filtered part of A . )

37

• Low - Rank Approximation of Order l
• A s1 u1 v1T s2 u2 v2T sp up
  vpT .
• Bl s1 u1 v1T s2 u2 v2T sl ul
  vlT
• Bl Ul
  Sl VlT
• Ul u1 , u2 , , ul ,
  
• Vl v1 , v2 , , vl ,
• Sl diag s1 , s2 , ,
  sl
• ( means computed values )

38

• What About Orthogonality ?
• Does UlT Ul I and VlT Vl I
  ?
• The theory behind the Power Method suggests
  that
• the more accurate are the computed singular
  triplets
• the smaller is the deviation from
  orthogonality .
• Is there a difference
• ( regarding deviation from orthogonality )
• between Ul and Vl ?

39

• Orthogonality Properties
• ( Assuming exact arithmetic . )
• Theorem 1 Consider the case when each
  singular
• triplet, sj , uj , vj , is computed
  by a finite
• number of "Left Iterations". ( At least
  one
• iteration for each triplet. ) In this case
• UlT Ul I and UlT Al
  0
• regardless the actual number of iterations !

40

• Left Iterations
• uk A vk-1 / vk-1T vk-1 ,
• vk AT uk / ukT uk .
• Right Iterations
• vk AT uk-1 / uk-1T
  uk-1 ,
• uk A vk / vkT vk .
• Can one see a difference?

41

• Orthogonality Properties
• ( Assuming exact arithmetic . )
• Theorem 2 Consider the case when each
  singular
• triplet, sj , uj , vj , is computed
  by a finite
• number of Right Iterations". ( At least
  one
• iteration for each triplet. ) In this case
• VlT Vl I and Al Vl 0
  
• regardless the actual number of iterations !

42

• Finite Termination
• Assuming exact arithmetic , r rank ( A
  ) .
• Corollary In both cases we have
• A Br s1 u1 v1T sr ur vrT
  ,
• regardless the number of iterations
• per singular triplet !

43

• A New QR Decomposion
• Assuming exact arithmetic , r rank ( A
  ) .
• In both cases we obtain an effective
• rank revealing QR decomposition
• A Ur Sr VrT .
• In Left Iterations UrT Ur I .
• In Right Iterations VrT Vr I .

44

• The Orthogonal Basis Problem
• Is to compute an orthogonal basis of Range ( A
  ).
• The Householder and Gram-Schmidt
• orthogonalizations methods use a
• column pivoting for size policy,
• which completely determine the basis.

45

• The Orthogonal Basis Problem
• The new method ,
• Orthogonalization via Deflation ,
• has larger freedom in choosing the basis.
  
• At the k th stage, the ultimate choice for a
• new vector to enter the basis is uk ,
• the k th left singular vector of A .
• ( But accurate computation of uk
• can be too expensive. )

46

• The Main Theme
• At the kth stage ,
• a few rectangular iterations
• are sufficient to provide
• a fair subtitute of uk .

47

• Applications in Missing Data Reconstruction
• Consider the case when some entries of A are
  missing.
• Missing Data in DNA Microarrays
• Tables of Annual Rain Data
• Tables of Water Levels in Observation
  Wells
• Web Search Engines
• Standard SVD algorithms are unable to handle
  such
• matrices.
• The Minimum Norm Approach is easily adapted
• to handle matrices with missing entries .

48

• A Modified Algorithm
• The objective function
• F ( u , v ) A - u vT
  F 2
• is redefined as
• F ( u , v ) S S ( aij ui
  vj ) 2 ,
• where the sum is restricted to known entries of
  A .
• ( As before,
• u (u1, u2, , um)T and v (v1, v2,
  , vn)T
• denote the vectors of unknowns. )

49

• The minimum norm approach
• Concluding Remarks
• Adds new insight into old methods and
  concepts.
• Fast Power methods. ( Relaxation methods,
  line search acceleration, etc. )
• Opens the door for new methods and concepts.
• ( The rectangular quotients equality,
  rectangular
• iterations, etc. )
• Orthogonalization via Deflation A new QR
• decomposition. ( Low - rank approximations,
• Rank revealing. )